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Chapter 8: Problem 15

Determine whether the quadratic expression is reducible. $$x^{2}+4 x+4$$

Short Answer

Expert verified
Yes, the quadratic expression \(x^{2}+4x+4\) is reducible and its factorised form is \((x + 2)^{2}\).

Step by step solution

01

Identify the Quadratic Expression

A quadratic expression is in the form \(ax^{2} + bx + c\). Here, the quadratic expression is \(x^{2}+4x+4\).
02

Check for Perfect Square Trinomial

Check if the given quadratic expression forms a perfect square trinomial. A perfect square trinomial fits the form \((a+b)^{2}= a^{2}+2ab+b^{2}\). Here, \(a^{2}=x^{2}\), \(2ab = 4x\) and \(b^{2} = 4\). We can conclude that \(a = x\) and \(b = 2\). Therefore, our expression is a perfect square trinomial.
03

Factorise the expression

Now, given that our quadratic expression is a perfect square trinomial, it can be further reduced as \((a + b)^{2}\). So, \(x^{2} + 4x + 4 = (x + 2)^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
The concept of a perfect square trinomial is an essential one in algebra. It is a quadratic expression that can be expressed in the form \((a + b)^2 = a^2 + 2ab + b^2\). Recognizing a perfect square trinomial helps simplify expressions and solve quadratic equations more easily.

For instance, in the quadratic expression \(x^2 + 4x + 4\), we need to check if it can be written as \((a+b)^2\). By comparing, we find \(a^2 = x^2\) and \(b^2 = 4\). The middle term, \(2ab = 4x\), confirms that it matches this perfect square form.

  • Identify \(a\) and \(b\) from \(a^2\) and \(b^2\).
  • Ensure the middle term equals \(2ab\).
  • Rewrite using \((a+b)^2\) if conditions are satisfied.
Identifying these patterns makes the process of factoring and solving quadratics much smoother.
Factoring Quadratics
Factoring quadratics is a fundamental skill for manipulating algebraic expressions, especially when solving quadratic equations. It involves rewriting a quadratic expression in a product of simpler expressions.

Starting with the expression \(x^2 + 4x + 4\), our goal is to factor it. For perfect square trinomials, like this one, terms are rearranged into squares. Recognizing \(x^2 + 4x + 4\) as \((x + 2)^2\) is crucial because once in this form, solving or simplifying equations becomes straightforward.

  • Look for patterns of perfect square trinomials.
  • Use the identity \((a+b)^2 = a^2 + 2ab + b^2\).
  • Rewrite the expression in its factored form.
This approach not only simplifies expressions but also aids in solving quadratic equations efficiently.
Algebraic Expressions
Algebraic expressions consist of variables, constants, and arithmetic operations. They form the building blocks in algebra and contain multiple components like terms and coefficients.

In an expression like \(x^2 + 4x + 4\), each part represents a specific component of the expression. Understanding these parts is key to manipulating and simplifying them.

  • Terms like \(x^2\), \(4x\), and \(4\) contribute to the whole expression.
  • The coefficient, such as \(4\) in \(4x\), multiplies the variable part.
  • Recognize patterns and apply identities for simplification.
Being comfortable with algebraic expressions allows for flexibility in transforming and solving complex mathematical problems.

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Most popular questions from this chapter

Find \(A^{2}\) (the product \(A A\) ) and \(A^{3}\) (the prod\(\left.u c t\left(A^{2}\right) A\right)\). $$A=\left[\begin{array}{rrr}3 & 0 & 0 \\\0 & 1 & 1 \\\\-4 & 1 & 0\end{array}\right]$$

Involve the use of matrix multiplication to transform one or more points. This technique, which can be applied to any set of points, is used extensively in computer graphics. Consider a series of points \(\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) such that, for every nonnegative integer \(i,\) the point \(\left(x_{i+1}, y_{i+1}\right)\) is found by applying the matrix \(\left[\begin{array}{cc}1 & -2 \\ 1 & -3\end{array}\right]\) tothe point \(\left(x_{i}, y_{i}\right)\) $$\left[\begin{array}{l}x_{i+1} \\\y_{i+1}\end{array}\right]=\left[\begin{array}{ll}1 & -2 \\\1&-3\end{array}\right]\left[\begin{array}{l}x_{i} \\\y_{i}\end{array}\right]$$ (a) Find \(\left(x_{1}, y_{1}\right)\) if \(\left(x_{0}, y_{0}\right)=(2,-1)\) (b) Find \(\left(x_{2}, y_{2}\right)\) if \(\left(x_{0}, y_{0}\right)=(4,6) .\) (Hint: Find \(\left(x_{1}, y_{1}\right)\) first.) (c) Use the inverse of an appropriate matrix to find \(\left(x_{0}, y_{0}\right)\) if \(\left(x_{3}, y_{3}\right)=(2,3)\)

A furniture manufacturer makes three different picces of furniture, each of which utilizes some combination of fabrics \(A, B,\) and \(C .\) The yardage of each fabric required for each piece of furniture is given in matrix \(F\). Fabric A Fabric B Fabric C (yd) \(\quad\) (yd) \(\quad\) (yd) \(\begin{array}{r}\text { Sofa } \\ \text { Loveseat } \\ \text { Chair }\end{array}\left[\begin{array}{ccc}10.5 & 2 & 1 \\ 8 & 1.5 & 1 \\ 4 & 1 & 0.5\end{array}\right]=F\) The cost of each fabric (in dollars per yard) is given in matrix \(C\).$$\begin{array}{l}\text { Fabric A } \\\\\text { Fabric B } \\\\\text { Fabric C }\end{array}\left[\begin{array}{r}10 \\\6 \\\5\end{array}\right]=C$$ Find the total cost of fabric for each piece of furniture.

Consider the following system of equations.$$\left\\{\begin{array}{r} x+y=3 \\\\-x+y=1 \\\2 x+y=6\end{array}\right.$$ Use Gauss-Jordan elimination to show that this system has no solution. Interpret your answer in terms of the graphs of the given equations.

A couple has \(\$ 10,000\) to invest for their child's wedding. Their accountant recommends placing at least \(\$ 6000\) in a high-yield investment and no more than \(\$ 4000\) in a low-yield investment. (a) Use \(x\) to denote the amount of money placed into the high-yield investment. Use \(y\) to denote the amount of money placed into the low-yield investment. Write a system of linear inequalities that describes the possible amounts the couple could invest in each type of venture. (b) Graph the region that represents all possible amounts the couple could put into each investment if they wish to follow the accountant's advice.
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